On A(P, N)-nuclearity and operator ideals By ESA NELIMAECKJU of Helsinki (Finland) (Eingegangen am 3.3. 1980) Introduction. In [8] RAMANUJAN and TERZIOQLU defined A,(&)-nuclear locally convex spaces associated with a power series space &(a), and they showed that many of the stability properties which are valid for nuclear locally convex spaces are valid for A,(a)-nuclear spaces, too. Replacing Am(&) by a K ~T H E space A(P) genarated by a stable, countable and monotone nuclear Gm-set P , RAMANUJAN and ROSENBERQER [9]
introduced the class of A(P, N)-nuclear locally convex spaces with similar permanence properties. It has been unknown whebher this class is defined by an operat,or ideal, i.e. whether it coincides with a class of %-spaces, in the sense of BETSCH [4], for some operator ideal 8. I n this paper we shall show that this is not the case in general and thus solve a problem of JARCHOW [7]. It will be shown that if the KOTHE space A(P) is not a power series space of infinite type, then the class of A(P, N)-nuclear spaces is not ideal generated. A similar result for AN(a)-nuclearity is obtained in the case lim inf oLn+I/an > 1 ; this will solve a problem proposed in [8].